@JoWehler The point is that while you can perhaps interpret the statement in an abstract way that the paper is only a carrier for, you can also interpret it directly as a physical property of the paper. Just as certain sheets of paper are solvable Sudoku puzzles and other sheets of paper are not, and certain Rubik's cubes are solvable and others are not, depending on the coloration of the faces. Would you agree that the solvability of a Rubik's cube is a physical property of the cube? In the same way that whether a lawnmower can be started is a physical property of the lawnmower.

@Peter-ReinstateMonica Then I ask you the same question. Do you agree that the solvability of a Rubik's cube or a Sudoku puzzle is a physical property of the Rubik's cube or Sudoku puzzle? The potential to write a proof of the Riemann's hypothesis on a sheet of paper is effectively nothing more than a very large and complex Sudoku puzzle. Sure, you can write a wrong equation, just as you can "solve" a Sudoku puzzle by breaking the rules or erasing numbers. But that doesn't affect whether the Sudoku puzzle is capable of being solved without breaking the rules.

@Peter-ReinstateMonica Well, nothing stops you from painting the Rubik's cube in the solved configuration. That you don't do that is a rule "in your mind" as well. But it is still a property of the Rubik's cube that it can be solved, or can't be solved, without doing things like that. Just as it is a property of the Sudoku that it can be solved, or can't be solved, without doing things like that.

@Peter-ReinstateMonica As a general principle, any physical property is specified by the circumstances under which that property shall be tested, and those circumstances are effectively only in the tester's mind. To measure the voltage in a circuit you have to hook up the voltmeter appropriately before reading the number - if you break the rule, then you just haven't measured the property. It is the same here; there are circumstances under which the solvability of the Sudoku is to be tested, and if you break the rules/change the circumstances, you just haven't correctly measured solvability.

"Just as certain sheets of paper are solvable Sudoku puzzles and other sheets of paper are not..." Doesn't whether a sheet of paper is a "solvable Sudoku puzzle" depend on the interpretation of that sheet of paper? I would posit it's the reading of the paper and construction of a mental model from it that makes it a Sudoku puzzle. If I came up with a new notation on paper that an appropriate Sudoku Puzzle Interpreter could read as a puzzle, and an appropriate Numeric Sequence Interpreter could read as π, which is it? What if you don't have either of these interpreters available?

@cjs The interpretation of a sheet of paper as a "solvable Sudoku puzzle" is captured by the phrase, "solvable Sudoku puzzle." Just as the interpretation of a circuit as having a current of 5 Amperes is captured by the phrase, "having a current of 5 Amperes." There is no fundamental difference between these two physical properties; in both cases there is an interpretation. All physical properties depend on an interpretation of the property, and are tested according to some test procedure in which the experimenter must follow certain rules.

Those are two very different things. "Is a solvable Sudoku puzzle" is information that may be encoded in many forms; I suppose one can have a debate about whether the information exists if nobody can interpret it. But a circuit with 5 A current flowing through it still has 5 A flowing through it even if nobody is observing or interpreting it.

Consider if I encrypt the solvable Sudoku puzzle information on that piece of paper. For someone with the key to decrypt it, it is indeed a solvable Sudoku puzzle. For someone without the key, it's random data. That's obviously not a Sudoku puzzle that they can solve. What if I have a machine generate a solvable Sudoku puzzle, write it encrypted under a random key, and then have the machine destroy the key? At that point nobody can solve that puzzle, and it is random data. Current flowing through a circuit does not behave at all like this.

@cjs "a circuit with 5 A current flowing through it still has 5 A flowing through it even if nobody is observing or interpreting it." - and a solvable Sudoku puzzle is still a solvable Sudoku puzzle even if nobody is observing or interpreting it. Most fundamentally, physical properties are statements of what would happen, if (subjunctively) a certain test were applied to the object. An encrypted Sudoku puzzle is not what we mean by a "solvable Sudoku puzzle" - it would be a different class of object. A Sudoku puzzle, normally understood, is a 9x9 grid with some numbers 1-9 in cells.

So I have two tests. 1. A person with the decryption key reads the paper; by that test the paper is a Sudoku puzzle. 2. A person without the decryption key reads the paper; by that test the paper is not a Sudoku puzzle. The same applies if I use a way of writing numbers that one person understands and another does not. Calling something like this that changes depending on who's observing it, and can be both X and not X at the same time to two different observers, is a funny way of defining a "physical property of an object," since the object itself is exactly the same in both cases.

@cjs An encrypted Sudoku puzzle is not what we mean by "a Sudoku puzzle," regardless of whether someone has the key.

if you just say "a Sudoku puzzle" this means an unencrypted puzzle, of the sort normally printed in books of Sudoku puzzles.

there is a test for whether something is "a Sudoku puzzle"; does it look like one? There isn't a test for whether something is "an encrypted Sudoku puzzle," because for the right key and encryption algorithm anything could be an encrypted Sudoku puzzle. There is a test for whether something is "an AES-256 encrypted Sudoku puzzle with key X" (assuming X is replaced with a specific key.)

I do not understand your distinction between an encrypted and an "unecrypted" Sudoku puzzle. Both are the same data, just in different forms, and both require an interpreter capable of reading the data to see the Sudoku puzzle within it. I use the encryption thing to make a clear distinction about who can and cannot read it, but it's the same thing if I happen to write out the puzzle using a set of glyphs for numbers that you don't understand.

@cjs If you write a Sudoku puzzle with a set of glyphs for numbers I don't understand, then again this is not what is meant when i just say "a Sudoku puzzle."

when I just say "a Sudoku puzzle," this phrase indicates a specific and fixed meaning. If someone has a different notion of what these three words might mean, then they're just talking about something else, and not talking about what I'm talking about.

it's like if I claim 2+2=4, this is in relation to a specific, fixed meaning of 2, +, =, and 4. If you're working in base 3 instead and say 2+2=1, then you haven't contradicted me and you're just talking past me.

So the physical properties of an object depend on what you understand about it, and for someone else the physical object may not have those physical properties?

no

@cjs when you say some words, what matters is what you mean by the words, and that meaning is the true statement you are making. This statement is independent of the interpreter, independent of you, independent of the words used to describe it.

Let's just stick with the piece of paper. You claim that whether or not a piece of paper is a Sudoku puzzle is a property of that physical object, yes?

when I say 2+2=4, what matters is what I mean by this. When I say a sheet of paper is "a Sudoku puzzle," what matters is what I mean by this. But once I've said it and meant it, I let it go; it is no longer tied to me or my words. It becomes objective.

yes, I do claim that

So when I claim that a piece of paper you can't understand is a Sudoku puzzle, than it objectively is?

you understand there is a difference between the words you say, and what they mean?

I don't get why you keep going back to "words" here. We're talking about physical properties of objects.

The meaning of my words is not connected to any physical objects, except inasmuch as you or I decide to connect them.

let's use subscripts. I claim the paper is not "a Sudoku puzzle"_1, where "a Sudoku puzzle"_1 denotes the meaning of the words "a Sudoku puzzle" when I used them. You, using a different interpretation, claim the paper is indeed "a Sudoku puzzle"_2, where "a Sudoku puzzle"_2 denotes the meaning of the words "a Sudoku puzzle" when you used them.

we are both right because we are talking about different things

that a sheet of paper is "a Sudoku puzzle"_1 is a physical property of the sheet of paper, which your sheet of paper lacks

even if I am dead, or in times before I was born, sheets of paper had or did not have this property of being "a Sudoku puzzle"_1

which is independent of the property of being "a Sudoku puzzle"_2

What do you mean "my" sheet of paper. We are both talking about the same sheet of paper.

the sheet of paper you're offering to me, that's not germane

It seems to me it is, unless you believe the physical properties of the object change for each observer.

what physical property changed? did you read anything I said?

Well, we have a property where for you the sheet of paper is not a Sudoku puzzle, and for me it is. That indicates to me that "is a Sudoku" puzzle is not a property of the sheet of paper itself, but is an interpretation of the sheet of paper.

stop saying "a Sudoku puzzle" please, because it's clear we mean different things by the term. Say either "a Sudoku puzzle"_1 or "a Sudoku puzzle"_2

for me the sheet of paper is not "a Sudoku puzzle"_1, and for you the sheet of paper is ALSO not "a Sudoku puzzle"_1. For you the sheet of paper is "a Sudoku puzzle"_2, and for me the sheet of paper is ALSO "a Sudoku puzzle"_2

each property is the same regardless of the observer

Ah, I see what you're getting at. I think.

So every "physical" property of an object depends on having an observer who can test it?

no, an object has a property if, were the test to be applied, a certain result would be obtained

this is regardless of whether the test is actually applied

But every test that could be applied has to be applied by someone, yes?

Or, maybe a different question will let us progress better: you seem to be claiming that all information carried by a physical object is a property of the object itself.

maybe, maybe not - maybe the test can be applied by an inanimate process

@cjs "information" is a slippery term, but sort of.

a property is "how an object causally relates to other objects"

or how it potentially would relate to other objects, given the opportunity

Ok, I think I see. So by your definition no object has inherent properties; it only has how it would react to something else. Everything is defined by its relationships with everything else.

So do you distinguish between properties that can have an "objective" test applied that will come out the same for all observers, versus properties that vary with the observer?

@cjs What properties vary with the observer? If there are such properties,

you would treat the observer as just another object

if a property of A depends on another object B in relation to A, then really it's not a property of A, it's a property of the pair (A, B)

like say, gravitational potential energy

Right. So "is a Sudoku puzzle"_1 is a property of the pair (paper, you), right?

no, it's a property of the paper

doesn't matter if I'm there or not

You don't need to be there, you just need to exist. Or potentially exist, right?

no

the paper has the property even if I never exist

I'm only the one to name this property, the property is there whether I name it or not, whether anyone does

So if you don't exist at all, thee paper still has an "is Sudoku"_1 property, that is "false" because you can't interpret it, even though you don't even exist?

@cjs no, it could have an "is Sudoku"_1 property that is true

@causative One property that varies with the observer is, "Is it a Sudoku puzzle?" Which I suppose you would say is a collection of properties, one for each person who could look at it, and an infinite number more for all the people who don't exist who could theoretically look at it if they do.

assuming it has the right arrangement of ink

Well, any arrangement of ink is right for some interpreters of it and wrong for others.

I wouldn't say it's one for each person who could look at it, I think most people share a similar "is a Sudoku puzzle"_1 property, involving certain assumptions like it being 9x9, it using Arabic numerals, etc.

it's not "for some interpreters"

the property is objective, just like, say, a sheet of paper being blue

if we can say a sheet of paper is blue, we can say it is half-blue and half-white, and that's equally objective

But "blue" is clearly not objective! Just go look at the blue/gold dress.

blue wavelengths

it can be made objective

and we can say that it is 99% white and 1% black, that's also objective

and we can specify where on the paper the black parts are

still objective

and we can specify a set of possible arrangements of the black parts - still objective

and that's all we need, to specify the set of all possible Sudoku puzzles

or, if you wish, to specify the set of all possible Sudoku_1 puzzles

"Blue wavelengths" is a very vague term that presupposes a particular kind of observer, and even there it's vague. You would make it an objective property by giving the wavelengths of light that are reflected by the object.

yes, that's how you would make it objective

I think the problem with this argument is that you can never confirm or deny that an object has property X, because if there's any disagreement you can say, "Oh, one person was really saying X1 and the other was really saying X2."

So rather than saying, "blue is not an objective property," you end up saying that it's really a set of properties, blue1, blue2, blue3, .... and you end up with any definition of blue anybody wants being an "objective" property of the object. Which pretty much destroys the meaning of "objective."

we can get partial agreement between people, that is, blue1 mostly intersects with blue2

and this is the basis of how we can communicate

we never quite get the full meaning the other person intends, but we can get most of it

But I think we do, in some cases, such as "this object is reflecting light of 475 nm wavelength."

in some cases, sure

although it's still probably not exactly the same meaning, because what's the measurement error in that 475 nm?

But it is the same meaning: just because I have a measurement error, known or unknown, doesn't change the property.

It's still reflecting 475 nm whether I measure it correctly or not.

(And it's understood, in science, that measurement errors exist. If I say it's 474 nm because I've made an error, that's an error, not a new "reflecting 475 nm"_2 property.)

when you say it's 475 nm, it's not going to be exactly and only 475 nm

maybe it's really 474.8544335109 nm

475 is too round a number to nail exactly

so when you say 475 nm and it's really 474.8544335109, you're not exactly wrong because the person you're speaking to understands there is some imprecision involved in the figure you give

You're confusing the measurement with the property of the object.

Just because I measure it wrong, doesn't change the object.

I'm here talking about an object that actually does reflect at 475 nm., regardless of our ability to measure it. You do agree that such a thing can exist, right?

well, it depends on the intended meaning. If when you say "it is reflecting 475 nm light," you really mean "it is reflecting 475 nm light +/- 1 nm," then you can be right even if it is 474.854435109 nm

and "it is reflecting 475 nm light +/- 1 nm" is a physical property of the object

But I don't intend that meaning. I'm saying it reflects at exactly 475 nm.

Do you agree that such an object can exist?

that's not physically possible because of Heisenberg's uncertainty principle

I think, anyway

no, yeah, I'm sure. if you know the exact wavelength, then you know the exact photon energy, which means you know the exact photon momentum, which means you cannot know the photon position at all

Ok, so the universe cannot have an object that reflects at exactly 475 nm?

well it could be possible in some other universe, maybe

but in this universe wavelength peaks are never at a perfect single frequency

How does Heisenberg's uncertainty principle apply here? That's about measurement, not the properties of objects.

it's not just about measurement, the Schrodinger equation is based on it

you can ask a physicist but I'm pretty sure I'm right

How so is it based on it?

well I am not a physicist so I can only be vague, but I believe quantum physics is based on the idea that the measurement error in the position of a particle is not just measurement error, it's a fundamental uncertainty represented by the wavefunction

Actually, I just went back and looked at your answer again, and I have another argument with it. When you say, "can this Sudoku be solved," it's not a property of the piece of paper because one could just as easily write it on a blackboard and solve it there. For any reasonable definition of "solving a Sudoku," it doesn't matter whether you solve it on the blackboard or the piece of paper. And that's math.

Essentially, "this solved Sudoku" is a property of the idea, independent of where and how you write it down.

I just asked a physicist on IRC and he said I'm right

about the emission spectra

anyway about what you are saying, it is the same that Jo Wehler said to me in the first comment

and my reply is the same: yes, you could interpret it that way, but you can also interpret it my way. There is a physical property of the ink-on-paper Sudoku, that it is solvable. It is not be the same as the property of ideas you are thinking of.

there is a set of possible arrangements of ink on paper, some of which are solvable Sudokus and some of which are not.

But the point is, given that you can interpret it my way, that shows that there is math that's not emperical.

I didn't, in my post, say otherwise

although privately I do believe all math is empirical

it's more complicated to explain your interpretation in physical terms, but it can be done

Ah. You should clarify you post then, by starting it with, "No; while some math can be empirical, not all of it is."

the bolded part of my answer is, "So, whether or not you wish to say that mathematics itself is empirical, you should admit that essentially every mathematical question corresponds to an empirical question"

that's clear enough

I don't commit to saying math itself is empirical or not

The arguments you make seem to lead toward, "everything is empirical," which makes "empirical" meaningless since it can't be used to distinguish anything.

Well, you should at least explain at the top that you're not answering the question asked.

I think I'm providing relevant context to the question asked, from which the reader may draw their own conclusions

Sure, but why be obscure about it? If you're not answering the question and just providing context, say so.

if the reader interprets it as an answer, and that the answer is that mathematics is empirical, I don't want to disabuse them of that, because that is indeed what I think

I don't explicitly say that math is empirical because it would be more of a challenge to support than the more narrow claim that I do support

math is empirical because all the activity of mathematicians is observing and acting upon physical systems - their own brains, each other's brains, pen and paper, computers

and there are beautiful abstract structures that arise because of this, and they arise because those structures are already present in physical reality

So in other words, you're giving people who expect they're reading an answer a bunch of text that is not an answer, but may lead them to share your conclusion that you can't support?

I support it as I just did

and it does count as an answer - it's not a direct yes or no, but it's something very closely relevant to what the OP was asking

And frankly, the idea that there are physical objects in the universe that embody the property that, e.g., there are more real numbers R than natural numbers N, seems, well, unlikely is the nicest way I can put it.

Actually, I don't even think your boldest statement is true. What is the empirical question corresponding to "are there more real numbers than natural numbers"?

@cjs It is whether it is possible to construct on paper an arrangement of ink following the rules of ZFC whose bottom line corresponds to the claim that there are more real numbers than natural numbers

provability is reducible to whether an artifact matching certain rigorous physical specifications can be constructed

and mathematics proceeds entirely through the construction of such artifacts. even informal mathematics is about constructing certain neural artifacts.

ultimately we know it has to be like this because everything a mathematician's brain is doing, it's only doing because of the neurons acting according to physical laws

so the results are the results of these physical laws

So then is there anything that is not empirical?

@cjs I don't think so. I am a panpsychist so I'd say all that is mind is also matter and subject to empirical investigation.